A random sample of n = 30 is drawn from a population that is normally distributed, and the sample variance is s©÷ = 41.5. Use ¥á = 0.05 in testing ¥Ç₀ : ¥ò©÷ = 29.0 versus ¥Ç©û : ¥ò©÷ ¡Á 29.0

The following is a reproduction of the post in Western encoding:

"A random sample of n = 30 is drawn from a population that is normally distributed, and the sample variance is s² = 41.5. Use α = 0.05 in testing Η₀ : σ² = 29.0 versus Η₁ : σ² ≠ 29.0"

s²/σ&sup2 = 41.5/29 = 1.431

The Chi-squared statistic for α = 0.95 and df=30-1=29 is 42.6.

Thus 95% of the samples of size N=30 would be such that
s²/σ&sup2 < 42.6/29=1.469

Since 1.431 < 1.469, we conclude that
Η₀ : σ² = 29.0 is accepted for α = 0.05.

Thank you MathMate

You're welcome.

To test the null hypothesis (H₀: σ² = 29.0) against the alternative hypothesis (H₁: σ² ≠ 29.0), you can use a chi-square test statistic. Here's how you can perform the test:

1. Define the hypotheses:
- Null Hypothesis (H₀): The population variance is 29.0 (σ² = 29.0)
- Alternative Hypothesis (H₁): The population variance is not equal to 29.0 (σ² ≠ 29.0)

2. Determine the significance level (α):
- The given significance level (α) is 0.05.

3. Calculate the test statistic:
- The chi-square test statistic (χ²) can be calculated using the formula:
χ² = (n - 1) * s² / σ²

- Given: Sample size (n) = 30, Sample variance (s²) = 41.5, Hypothesized population variance (σ²) = 29.0

- Substitute these values into the formula:
χ² = (30 - 1) * 41.5 / 29.0
χ² ≈ 43.09

4. Determine the critical value(s):
- Since the alternative hypothesis is two-tailed (σ² ≠ 29.0), you need to find the critical values in both tails of the chi-square distribution.

- The chi-square distribution with (n - 1) degrees of freedom is used here.

- Find the critical values (χ²₁ and χ²₂) such that P(χ² < χ²₁) = α/2 and P(χ² > χ²₂) = α/2.

- Using a chi-square table or calculator at a significance level of α = 0.05 and (n - 1) = 29 degrees of freedom, you can find the critical values to be approximately 16.05 and 45.72.

5. Make a decision:
- Compare the test statistic (χ²) to the critical values (χ²₁ and χ²₂) to make a decision.

- If χ² < χ²₁ or χ² > χ²₂, reject the null hypothesis (H₀).

- If χ²₁ < χ² < χ²₂, fail to reject the null hypothesis (H₀).

6. State the conclusion:
- Based on the calculations and comparison, if χ² < χ²₁ or χ² > χ²₂, you would reject the null hypothesis (H₀) and conclude that there is sufficient evidence to suggest that the population variance is not equal to 29.0 (σ² ≠ 29.0).

Remember that in hypothesis testing, the decision-making process relies on comparing the calculated test statistic to the critical values at the given significance level.